Holomorphic Poincaré conjecture

Question

Is there a notion of n-sphere for complex manifolds so that one could formulate a Holomorphic Poincaré conjecture ?

Answer 1

It is known that the only spheres which can possibly be complex manifolds are $S^2$ and $S^6$. Of course $S^2$ is known to admit an integrable complex structure as the Riemann sphere, but it is still an open problem whether $S^6$ admits an integrable complex structure. By the uniformization theorem, $S^2$ admits a unique complex manifold structure.

See the theorem on page 212 of May's A Concise Course in Algebraic Topology for a proof using characteristic classes that $S^2$ and $S^6$ are the only spheres that even admit an almost complex structure. The notes are freely available on his website here.